Problem 16 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{1+\frac{4}{x}+\frac{4}{x^{2}}}{1-\frac{2}{x}-\frac{8}{x^{2}}}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{1+\frac{4}{x}+\frac{4}{x^{2}}}{1-\frac{2}{x}-\frac{8}{x^{2}}}\) is \(\frac{x+2}{x-4}\).

Step 1: Multiply numerator and denominator by \(x^2\)

Multiply both the numerator and the denominator by \(x^2\), which is the least common multiple (lcm) of \(x\), \(x^2\). This step will help remove the fractions within fractions. The equation will become \[\frac{x^2+4x+4}{x^2-2x-8}\]

Step 2: Factorize the numerator and the denominator

Next, factorize the numerator and the denominator separately. The numerator can be expressed as the square of a binomial, \(x+2\), and the denominator can be expressed as \((x-4)(x+2)\). The expression will become \[\frac{(x+2)^2}{(x-4)(x+2)}\] Source:

Step 3: Cancel common factors

In the expression \((x+2)^2 / ((x-4)(x+2))\), \((x+2)\) is a common factor to both the numerator and denominator. So, \((x+2)\) can be cancelled from both the numerator and denominator. The simplified expression will be \[\frac{x+2}{x-4}.\]

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in simplifying algebraic fractions. It involves breaking down a polynomial into simpler components, called factors, that when multiplied together give back the original polynomial. This is similar to factoring a number into its prime components.

For example, the polynomial equation \(x^2 + 4x + 4\) can be factored by looking for two numbers that add up to the middle coefficient (4 in this case) and multiply to the last constant term (also 4). We find that 2 and 2 fit the bill, leading us to the factored form \( (x+2)(x+2) \) or \( (x+2)^2 \).

Factoring becomes particularly useful when simplifying algebraic fractions, as it may reveal common factors in the numerator and denominator that can be canceled to simplify the expression. To successfully factor a polynomial, one must understand various methods such as finding the greatest common factor (GCF), using the difference of squares, or applying the quadratic formula when necessary.

Least Common Multiple

The least common multiple (LCM) of two or more algebraic expressions is the smallest expression that is a multiple of each of the expressions. Finding the LCM is a crucial step in simplifying complex fractions where variables have different exponents or when combining fractions with different denominators.

In our exercise, the LCM of \( x \) and \( x^2 \) was utilized to clear out the fractional parts by multiplying both the numerator and denominator by \( x^2 \)—the smallest power of \( x \) that both denominators divide into evenly. This process standardized the denominator across all terms, allowing us to combine and simplify the expression as a whole.

Understanding how to find the LCM can significantly simplify mathematical operations involving fractions by creating a common denominator. This enables the addition, subtraction, or comparison of fractions with ease.

Canceling Common Factors

Canceling common factors in an algebraic fraction simplifies the expression to its lowest terms. After factoring the numerator and the denominator, we can look for any terms that appear in both and eliminate them.

Consider our equation \(\frac{(x+2)^2}{(x-4)(x+2)}\). Here, \(x+2\) occurs in both the numerator and denominator. Since any number divided by itself equals one, we can cancel out the common factor \(x+2\). It's essential to note that we can only cancel factors, not terms that are added or subtracted within an expression.

By doing so, we are left with \(\frac{x+2}{x-4}\), a much simpler expression. Removing common factors is a powerful tool in algebra that often leads to more manageable and understandable results, making it an invaluable step in solving algebraic problems.